We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the V-pro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors F: L ✲ K induce equipment arrows relF: relL ✲ relK, spnF: spnL ✲ spnK, and so on, and similarly for arbitrary monoidal functors V ✲ W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A.Joyal and R.Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category. In this paper we extend this argument to arbitrary dimension. We demonstrate that for an n-operad A in the author’s sense there exists a symmetric operad S n (A) called the n-fold suspension of A such that the

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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higher-dimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences

Preface to the reprinted edition Soon after the appearance of enriched category theory in the sense of Eilenberg-Kelly1, I wondered whether V-categories could be the same as W-categories for non-equivalent monoidal categories V and W. It was not until my four-month sabbatical in Milan at the end of 1981 that I made a serious attempt to properly formulate this question and try to solve it. By this time I was very impressed by the work of Bob Walters [28] showing that sheaves on a site were enriched categories. On sabbatical at Wesleyan University (Middletown) in 1976-77, I had looked at a preprint of Denis Higgs showing that sheaves on a Heyting algebra H couldbeviewedassomekindofH-valued sets. The latter seemed to be understandable as enriched categories without identities. Walters ’ deeper explanation was that they were enriched categories (with identities) except that the base was not H but rather a bicategory built from H. A stream of research was initiated in which the base monoidal category for enrichment was replaced, more generally, by a base bicategory. In analysis, Cauchy complete metric spaces are often studied as completions of more readily defined metric spaces. Bill Lawvere [15] had found that Cauchy completeness could be expressed for general enriched categories with metric spaces as a special case. Cauchy sequences became left adjoint modules2 and convergence became representability. In Walters ’ work it was the Cauchy complete enriched categories that were the sheaves. It was natural then to ask, rather than my original question, whether Cauchy complete V-categories were the same as Cauchy complete W-categories for appropriate base bicategories V and W. I knew already [20] that the bicategory V-Mod whose morphisms were modules between V-categories could be constructed from the bicategory whose morphisms were V-functors. So the question became: given a base bicategory V, for which